
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear response functions: \Sec{CCLR}}\label{sec:cclr}
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In the \Sec{CCLR  } section the input that is
specific for coupled cluster linear response properties is read in. 
This section includes presently 
\begin{itemize}
\item frequency-dependent linear response properties 
      $\alpha_{AB}(\omega)  = - \langle\langle A; B \rangle\rangle_\omega$
      where $A$ and $B$ can be any of the one-electron
      operators for which integrals are available in the 
      \Sec{*INTEGRALS} input part.
\item dispersion coefficients $D_{AB}(n)$ for $\alpha_{AB}(\omega)$
      which for $n \ge 0$ are defined by the expansion
      $$ \alpha_{AB}(\omega) = \sum_{n=0}^{\infty} \omega^n \, D_{AB}(n) $$
      In addition to the dispersion coefficients for $n \ge 0$
      there are also coefficients available for $ n = -1, \ldots, -4$,
      which are related to the Cauchy moments by $ D_{AB}(n) = S_{AB}(-n-2)$.
      \\
      Note, that for real response functions only even moments
      $D_{AB}(2n) = S_{AB}(-2n-2)$ with $n \ge -2$ are available,
      while for imaginary response functions only odd moments
      $D_{AB}(2n+1) = S_{AB}(-2n-3)$ with $n \ge -2$ are available.
\end{itemize}
Coupled cluster linear response functions and dispersion coefficients
are implemented for the models CCS, CC2 and CCSD. 
Publications that report results obtained with CC linear response
calculations should cite Ref.\ \cite{Christiansen:CCLR}. 
For dispersion coefficients also a citation of Ref.\ \cite{Haettig:CAUCHY} 
should be included.

\begin{description}
% \item[\Key{RELAXE}] 
%
% \item[\Key{UNRELA}] 
%
\item[\Key{FREQUE}] \verb| |\newline
READ (LUCMD,*) NBLRFR
READ (LUCMD,*) (BLRFR(I),I=1,NBLRFR)

Frequency input for $\langle\langle A;B \rangle\rangle_{\omega}$.
 
\item[\Key{DIPOLE}] 
Evaluate all symmetry allowed elements of the dipole polarizability
(max. 6 components).
%
%\item[\Key{ALLDSP}] 
%
% \item[\Key{FTST  }] 
%
\item[\Key{ASYMSD}] 
Use an asymmetric formulation of the linear response function which
does not require the solution of response equations for the operators $A$, 
but solves two sets of response equations for the operators $B$.
%
\item[\Key{DISPCF}] \verb| |\newline
   \verb|READ (LUCMD,*) NLRDSPE|

   Calculate the dispersion coefficients 
   $D_{AB}(n)$ up to $n = $ \verb+NLRDSPE+.
%
\item[\Key{OPERAT}] \verb| |\newline
   \verb|READ (LUCMD,'(2A)') LABELA, LABELB|\newline
   \verb|DO WHILE (LABELA(1:1).NE.'.' .AND. LABELA(1:1).NE.'*')|\newline
   \verb|  READ (LUCMD,'(2A)') LABELA, LABELB|\newline
   \verb|END DO|

Read pairs of operator labels. 
 
\item[\Key{AVERAG}] \verb| |\newline
   \verb|READ (LUCMD,'(A)') AVERAGE|\newline
   \verb|READ (LUCMD,'(A)') SYMMETRY|

Evaluate special tensor averages of linear response functions.
Presently implemented are the isotropic average of the dipole polarizability
$\bar{\alpha}$ and the dipole polarizability anisotropy $\alpha_{ani}$.
Specify \verb+ALPHA_ISO+ for \verb+AVERAGE+ to obtain $\bar{\alpha}$ and
\verb+ALPHA_ANI+ to obtain $\alpha_{ani}$ and $\bar{\alpha}$.
The \verb+SYMMETRY+ input defines the selection rules that can be
exploited to reduce the number of tensor elements that have to be
evaluated. Available options are
\verb+ATOM+, \verb+SPHTOP+ (spherical top), \verb+LINEAR+,
\verb+XYDEGN+ ($x$- and $y$-axis equivalent, i.e.\ a $C_z^n$
symmetry axis with $n \ge 3$),  and \verb+GENER+ (use point
group symmetry from geometry input).
 
\item[\Key{PRINT }] \verb| |\newline
   \verb|READ (LUCMD,*) IPRSOP|

   Set print level for linear response output.
 
\end{description}
